A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case

TL;DR

This paper constructs a Lyapunov functional for perturbed semilinear wave equations in the critical case, demonstrating that blow-up rates match those of the unperturbed ODE using advanced inequalities.

Contribution

It introduces a Lyapunov functional in similarity variables for perturbed critical wave equations, linking blow-up behavior to the unperturbed ODE.

Findings

Blow-up rate matches the solution of the associated ODE

Established a Lyapunov functional for the perturbed problem

Used interpolation and Gagliardo-Nirenberg inequality in analysis

Abstract

We consider in this paper some class of perturbation for the semilinear wave equation with critical (in the conformal transform sense) power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using a two-step argument based on interpolation and a critical Gagliardo-Nirenberg inequality, we show that the blow-up rate of any sigular solution is given by the solution of the non perturbed associated ODE, namely $u" = u^p$.